Hereditarily countable set

In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets.

The inductive definition above is well-founded and can be expressed in the language of first-order set theory.

A set is hereditarily countable if and only if it is countable, and every element of its transitive closure is countable.¹

The class of all hereditarily countable sets can be proven to be a set from the axioms of Zermelo–Fraenkel set theory (ZF) and is set is designated ${\displaystyle H_{\aleph _{1}}}$. In particular, the existence does not require any form of the axiom of choice. Constructive Zermelo–Fraenkel (CZF) does not prove the class to be a set.

The set ${\displaystyle H_{\aleph _{1}}}$ is included in the set ${\displaystyle V_{\omega _{1}}}$ from the von Neumann hierarchy, that is, ${\displaystyle H_{\aleph _{1}}\subseteq V_{\omega _{1}}}$. Every hereditarily finite set is hereditarily countable, so ${\displaystyle H_{\aleph _{1}}\supseteq \mathrm {HF} =V_{\omega }}$. Since ${\displaystyle V_{\omega }}$ is countable, we in fact have ${\displaystyle H_{\aleph _{1}}\supseteq V_{\omega +1}}$.

An ordinal is hereditarily countable if and only if it is countable.

This class is a model of Kripke–Platek set theory with the axiom of infinity (KPI), if the axiom of countable choice is assumed in the metatheory.

If ${\displaystyle x\in H_{\aleph _{1}}}$, then ${\displaystyle L_{\omega _{1}}(x)\subseteq H_{\aleph _{1}}}$.

More generally, a set is hereditarily of cardinality less than κ if it is of cardinality less than κ, and all its elements are hereditarily of cardinality less than κ. The class of all such sets can also be proven to be a set from the axioms of ZF, and is designated ${\displaystyle H_{\kappa }\!}$. If the axiom of choice holds and the cardinal κ is regular, then a set is hereditarily of cardinality less than κ if and only if its transitive closure is of cardinality less than κ.